However, the moduli space \(\mathrm {PT}_{\mathcal {X}/S}^{(P)}\)  with its virtual fundamental class \([\mathrm {PT}_{\mathcal {X}/S}^{(P)}]^\mathrm {vir}\)  is not the right space to define directly relative orbifold PT invariants and hence
to derive a degeneration formula of orbifold PT theory, but it provides an elementary basis to construct the desired good degeneration of moduli spaces of PT stable pairs, see Section 5.
Next, we will deal with the orbifold PT side parallel to the DT theory in [[1]}, [2]}]. We briefly describe the construction of this good degeneration as follows. Now, let \(\pi :\mathcal {X}\rightarrow \mathbb {A}^1\)  be a simple degeneration of relative dimension three (see Definition REF ) and also a family of projective Deligne-Mumford stacks with a coarse moduli scheme \(X\)  and a relative polarization \((\mathcal {E}_{\mathcal {X}},\mathcal {O}_{X}(1))\) . Following the method in [[3]}, [1]}], a good degeneration constructed here is to fill some suitable central fiber in the following family
\(\coprod _{c\ne 0}\mathrm {PT}_{\mathcal {X}_{c}/\mathbb {C}}^{P},\) 
